50.7.2 problem 1(b)

Internal problem ID [7906]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.9. Reduction of Order. Page 38
Problem number : 1(b)
Date solved : Sunday, March 30, 2025 at 12:36:46 PM
CAS classification : [NONE]

\begin{align*} x y y^{\prime \prime }&=y^{\prime }+{y^{\prime }}^{3} \end{align*}

✗ Maple

ode:=x*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)+diff(y(x),x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 1.783 (sec). Leaf size: 103

ode=x*D[y[x],{x,2}]==D[y[x],x]+(D[y[x],x])^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_2-i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2} \\ y(x)\to i e^{-c_1} \sqrt {-1+e^{2 c_1} x^2}+c_2 \\ y(x)\to c_2-i \sqrt {x^2} \\ y(x)\to i \sqrt {x^2}+c_2 \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**3 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (-27*x*y(x)*Derivative(y(x), (x, 2))/2 + sqrt(729*x**2*y(x)**2*Derivative(y(x), (x, 2))**2 + 108)/2)**(1/3)/3 + Derivative(y(x), x) - 1/(-27*x*y(x)*Derivative(y(x), (x, 2))/2 + sqrt(729*x**2*y(x)**2*Derivative(y(x), (x, 2))**2 + 108)/2)**(1/3) cannot be solved by the factorable group method